The process of crystallization of metal alloys and many associated structural patterns of alloys are described using phase equilibrium diagrams. These diagrams show in convenient graphical form the phase composition and structure as a function of temperature and concentration. The diagrams are plotted for equilibrium conditions; the equilibrium state corresponds to the minimum free energy value.
Examination of phase diagrams allows one to determine phase transformations under conditions of very slow cooling or heating. The pattern of changes in the number of phases in a heterogeneous system is determined by the phase rule.
Phase- a homogeneous part of the system, separated from other parts of the system (phases) by an interface, during the transition through which the chemical composition or structure of the substance changes abruptly.
When studying physicochemical equilibria, temperature and pressure are taken as external factors affecting the state of the alloy. Applying the phase rule to metals, in many cases it is possible to accept only one external factor as changing - temperature, because pressure, except for very high pressure, has little effect on the phase equilibrium in the solid and liquid states. Then the general laws of the existence of stable phases that meet the equilibrium conditions are expressed in mathematical form by the phase rule (Gibbs rule) and at constant pressure is expressed by the following equation:
C = K + 1 - ,
Where TO– number of components in the system; - number of phases; WITH– number of degrees of freedom (system variability).
The number of degrees of freedom C is the number of independent internal variables (phase composition) and external (temperature, pressure) factors that can be changed without changing the number of phases in equilibrium. During phase transformations in alloys, the newly formed phase does not necessarily have to have a lower level of free energy than the original one, but the free energy of the system as a whole must necessarily decrease during the process of phase transformation.
From free energy curves, the main types of phase diagrams can be constructed geometrically. They are plotted in temperature-concentration coordinates as a percentage by mass.
To construct phase diagrams, thermal analysis developed by N.S. is used. Kurnakov. Cooling curves of individual alloys are experimentally obtained and, based on their bends or stops associated with the thermal effects of transformations, the temperature of the corresponding transformations is determined. These temperatures are called critical points.
When studying transformations in the solid state, various methods of physicochemical analysis, microanalysis, X-ray diffraction, dilatometric, magnetic analysis, etc. are used.
In the liquid state, most metals dissolve indefinitely in one another, forming a single-phase liquid solution. Any phases formed in the alloy differ in composition from the original liquid solution. Therefore, for the formation of a stable embryo, not only heterogeneous fluctuations are necessary, but also concentration fluctuations. Concentration fluctuations are temporary deviations in the chemical composition of an alloy in individual small volumes of a liquid solution from its average composition. Such fluctuations arise as a result of the diffusion movement of atoms of a substance and due to thermal movements in a liquid solution. The nucleus of a new phase can arise only in those microvolumes of the initial phase, the composition of which, as a result of fluctuations in the concentration and arrangement of atoms, corresponds to the composition and structure of the new crystallizing phase.
The rate of crystal growth in liquid solutions is less than in pure metals. This is explained by the fact that crystal growth requires the diffusion movement of component atoms in a liquid solution.
The state diagram is divided into areas by lines. Some areas may consist of only one phase, and some may consist of two phases, having different compositions, structures and properties.
By analyzing the phase diagram, one can get an idea of the specific properties of the alloys of a given system of components and the nature of their changes depending on the composition, as well as the possibility of heat treatment of the alloys and the heating temperature for it.
The type of diagram is determined by the nature of the interactions that arise between the components in the liquid and solid states.
Phase diagram for alloys forming mixtures of pure components
Both components of the alloy are infinitely soluble in the liquid state, and insoluble in the solid state and do not form chemical compounds and do not have polymorphic transformations. General view of the diagram in Fig. 3. Phases: liquid - L, crystals - A and B.
Line DIA– the line of the beginning of crystallization is the line liquidus; line DSE– the line of the end of crystallization is the line solidus. On line AC crystals begin to stand out A; on line NE– crystals IN; on line DSE from liquid concentration WITH crystals are released at the same time A And IN. A eutectic mixture of two types of crystals simultaneously crystallizing from a liquid is called eutectic.
Rice. 3. General view of the state diagram and cooling curves of the alloys: 1 – hypereutectic; 2 – hypoeutectic; 3 – eutectic.
In Fig. Figure 4 schematically shows the structure of the alloy at different moments of crystallization.
Rice. 4. Structure of alloys
Having a phase diagram, you can trace the phase transformations of any alloy and indicate the composition and quantitative ratio of the phases at any temperature. This is determined by the rule of segments.
To determine the concentration of components in two phases through a given point A(Fig. 3.), characterizing the state of the alloy, draw a horizontal line until it intersects with the lines limiting this area. Projections of intersection points V And With phase compositions will be shown on the horizontal axis of the diagram V 1 And With 1 . The segments of this line between the point A and dots V And With, which determine the compositions of the phases, are inversely proportional to the quantities of these phases:
AND=ac/bc; B=ab/bc.
These rules are valid for any two-phase region of the phase diagram.
When assessing the strength and other properties of the alloy, it should be borne in mind that the part of the alloy that is represented by the eutectic has a higher strength than the part represented by larger grains of the excess phase.
Phase diagram for alloys with unlimited solubility in the solid state
In Fig. Figure 5 shows a phase diagram for alloys with unlimited solubility of components in each other in the liquid and solid states, having the same types of lattices and a similar structure of the outer electron shells.
Line AMV– line liquidus; line ANIN– line solidus; phase is a solid solution of components A And IN, the grains of this phase have a single crystal lattice, but in alloys of different compositions the number of component atoms A And IN in the unit cells of the lattice is different.
Crystallization of α-phases in alloys of various kinds occurs in accordance with the rule of segments. In the case of equilibrium crystallization, which occurs at a sufficiently low cooling rate of the alloy, by the end of crystallization the composition of the finally formed phase A 4 must correspond to the original alloy composition IN 1 (in this case alloy I). This is due to the continuously occurring diffusion between both phases.
Rice. 5. General view of the state diagram and cooling curve of the alloy.
In the case of accelerated cooling of the alloy during crystallization, diffusion processes do not have time to complete. In this regard, the central part of each grain turns out to be enriched with a more refractory component IN, and the peripheral one is a low-melting component A. This phenomenon is called dendritic liquation, reducing the strength and other properties of alloys.
Dendritic segregation can be eliminated by prolonged annealing. This annealing is called diffusion annealing. The diffusion processes that occur in this case level out the chemical composition in the grains.
When a solid solution is formed, the tensile strength, yield strength and hardness increase while maintaining a sufficiently high ductility. This is explained by the fact that the atoms of the dissolved element are grouped in distorted regions of the lattice, which interferes with the advancement of dislocations.
PHASE DIAGRAM, a graphic representation of the conditions (temperature, pressure, chemical composition, etc.) under which in an equilibrium thermodynamic system consisting of one or more given substances (independent components of the system), homogeneous states of matter (phases) exist with differing physicochemical properties. The term “state diagram” is used as equivalent to the term “phase diagram” (mainly in Russia and Germany). However, a phase diagram is often, especially in the English-language literature, also called graphs that do not directly reflect phase equilibria in the system.
Phases are represented on a phase diagram in the form of regions bounded by curves or surfaces located in the space of independent thermodynamic variables. Usually this is temperature T, pressure P, mole fractions of components of the system x, functions of these and other variables, such as ratios of quantities or concentrations of components, densities p or molar volumes V m, partial pressures or chemical potentials of substances μ. In the absence of external force fields, the number of coordinate axes of the complete phase diagram of an open system with components is equal to c+2. To depict multidimensional phase diagrams on a plane, their sections and projections are used, constructed under certain restrictions imposed on some of the independent variables, often in combination with specially selected coordinate systems (Jenicke coordinates, Gibbs-Rosebohm triangles, etc.). The phase diagram shows: what individual substances, liquid, solid or gas solutions form the given components of the system; under what conditions are such phases and their heterogeneous mixtures thermodynamically stable; at what values of thermodynamic variables do phase transformations of substances occur in the system. Phase diagrams, containing data on the chemical composition of phases, also make it possible to determine the relative amounts of coexisting phases. Such information is necessary for solving many scientific and practical problems and is widely used in chemistry, metallurgy, materials science, geochemistry and other fields of science and technology.
The coordinates of the phase diagram can be thermodynamic variables of two types - parameters of thermal, mechanical and chemical equilibrium T, P, μ, which have the same values in all parts of the equilibrium system, or (usually different in different phases) generalized densities of extensive properties, such as x, p , V m and other properties equal to the ratio of extensive quantities to the amount, mass or volume of matter in the system. In this regard, three types of phase diagrams are distinguished. Diagrams of the same type are isomorphic: they have the same topological features regardless of the number of components and the values of specific variables on the coordinate axes.
In phase diagrams of the type (T, P), (T, μ i), (μ i, μ j) and the like, with intensive equilibrium parameters, only phase regions and the lines (surfaces) separating them are presented, which indicate the boundaries of stability of individual phases The points of intersection of the lines correspond to the equilibrium conditions of more than two phases. Thus, the triple point on the (T, P) diagram of a one-component system indicates the conditions for the stable coexistence of three phases.
If there is an axis of concentrations, molar properties, densities, as, for example, in the phase diagram (T, x), (P, x), (μ i, x), (T, p), regions of phase stability are separated by other regions that reflect the existence of heterogeneous mixtures of equilibrium phases. A phase diagram of this type for a two-component cadmium-zinc system is shown in the figure. The diagram of this two-component system has two coordinate axes, and not four, as required by the above expression with + 2, since in its construction the condition of constancy of P was used and two independent variables of the amount of Cd and Zn were replaced by one concentration x Zn (x Cd = 1 - x Zn). The upper part of the figure is a liquid-vapor equilibrium diagram. The broken curve connecting the melting points of pure components is called the liquidus line; it shows the “fusibility diagram” of the system. Straight lines (conodes) drawn in the heterogeneous region of such a phase diagram between the boundaries of two coexisting phases parallel to the concentration axis (see eutectic conode in the figure) allow, for any given component composition of the system, to determine the quantities of coexisting phases (“leverage rule”).
In the phase diagram of the third type - (x i, x j), (x i, p), (molar entropy, x), (molar enthalpy, x), etc. - the coordinates are only generalized densities of extensive thermodynamic properties. These diagrams also represent heterogeneous mixtures of phases and conodes, but, unlike the other two types of phase diagrams, in this case the state of heterogeneous mixtures is displayed by a flat or volumetric figure (triangle, tetrahedron) and it is possible to determine the quantitative phase composition of the system in the equilibrium of three and more phases (“rule of the center of gravity” of the figure).
Phase diagrams are studied experimentally and calculated using chemical thermodynamics methods based on data on the thermodynamic properties of the substances that make up the system. The theoretical basis for constructing phase diagrams was given by J. Gibbs in the 1880s. He also formulated the “phase rule” (see Gibbs phase rule), which is widely used in the experimental study of phase equilibria and phase diagrams: for fixed T and P, the number of equilibrium coexisting phases f cannot exceed the number of components of the system by more than two, f ≤ c + 2.
Lit.: Palatnik L. S., Landau A. I. Phase equilibria in multicomponent systems. Har., 1961; Kaufman L., Bernstein H. Calculation of state diagrams [of metallic systems] using a computer. M., 1972; Physical metallurgy / Edited by R. Kahn, P. Haazen. M., 1987. T. 2.
Figure 3.3 shows the phase diagram in P–V coordinates, and Figure 3.4 shows the phase diagram in T–S coordinates.
Fig.3.3. Phase P-V diagram Fig. 3.4. Phase T-S diagram
Designations:
t + l – region of equilibrium coexistence of solid and liquid
t + p – region of equilibrium coexistence of solid and vapor
l + n – region of equilibrium coexistence of liquid and vapor
If on the P–T diagram the areas of two-phase states were depicted as curves, then on the P–V and T–S diagrams these are some areas.
Line AKF is called the boundary curve. It, in turn, is divided into a lower boundary curve (section AK) and an upper boundary curve (section KF).
In Figs. 3.3 and 3.4, line BF, where the regions of three two-phase states meet, is the extended triple point T from Figs. 3.1 and 3.2.
When a substance melts, which, like vaporization, occurs at a constant temperature, an equilibrium two-phase mixture of solid and liquid phases is formed. The values of the specific volume of the liquid phase in the composition of a two-phase mixture are taken in Fig. 3.3 from the AN curve, and the values of the specific volume of the solid phase - from the BE curve.
Inside the area limited by the AKF contour, the substance is a mixture of two phases: boiling liquid (L) and dry saturated vapor (P).
Due to the additivity of volume, the specific volume of such a two-phase mixture is determined by the formula
specific entropy:
Singular points of phase diagrams
Triple point
The triple point is the point at which the equilibrium curves of the three phases converge. In Fig. 3.1 and 3.2 this is point T.
Some pure substances, for example, sulfur, carbon, etc., in the solid state of aggregation have several phases (modifications).
There are no modifications in liquid and gaseous states.
In accordance with equation (1.3), in a one-component thermal deformation system, no more than three phases can simultaneously be in equilibrium.
If a substance has several modifications in the solid state, then the total number of phases of the substance exceeds three and such a substance must have several triple points. As an example, Fig. 3.5 shows the P–T phase diagram of a substance that has two modifications in the solid state of aggregation.
Fig.3.5. Phase P-T diagram
substances with two crystalline
what phases
Designations:
I – liquid phase;
II – gaseous phase;
III 1 and III 2 – modifications in the solid state of aggregation
(crystalline phases)
At the triple point T 1, the following are in equilibrium: gaseous, liquid and crystalline phases III 2. This point is basic triple point.
At the triple point T2, the following are in equilibrium: liquid and two crystalline phases.
At the triple point T3, the gaseous and two crystalline phases are in equilibrium.
There are five known crystalline modifications (phases) of water: III 1, III 2, III 3, III 5, III 6.
Ordinary ice is the crystalline phase III 1, and other modifications are formed at very high pressures of thousands of MPa.
Ordinary ice exists up to a pressure of 204.7 MPa and a temperature of 22 0 C.
The remaining modifications (phases) are ice denser than water. One of these ices, “hot ice,” was observed at a pressure of 2000 MPa up to a temperature of + 80 0 C.
Thermodynamic parameters basic triple point of water the following:
T tr = 273.16 K = 0.01 0 C;
P tr = 610.8 Pa;
Vtr = 0.001 m 3 /kg.
Melting curve anomaly ( 
) exists only for regular ice.
Introduction
Phase diagrams are an integral part of any discussion of material properties when we are talking about the interaction of different materials. Phase diagrams are especially important in microelectronics, because For the manufacture of leads and passivation layers, a large range of different materials must be used. In the production of integrated circuits, silicon is in close contact with various metals; we will pay special attention to those phase diagrams in which silicon appears as one of the components.
This abstract discusses what types of phase diagrams there are, the concept of phase transition, solid solubility, and the most important systems of substances for microelectronics.
Types of phase diagrams
Single-phase phase diagrams are graphs that depict the phase state of only one material depending on pressure, volume and temperature. It is usually not customary to draw a three-dimensional graph on a two-dimensional plane - they depict its projection onto the temperature - pressure plane. An example of a single-phase state diagram is given in Fig. 1.
Rice. 1.
The diagram clearly demarcates the areas in which the material can exist in only one phase state - as a solid, liquid or gas. Along the demarcated lines, a substance can have two phase states (two phases) that are in context with each other. Any of the combinations takes place: solid - liquid, solid - vapor, liquid - vapor. At the point where the lines of the diagram intersect, the so-called triple point, all three phases can exist simultaneously. Moreover, this is possible at one single temperature, so the triple point serves as a good reference point for temperatures. Typically, the reference point is the triple point of water (for example, in precision measurements using thermocouples, where the reference junction is in contact with the ice-water-steam system).
A binary phase diagram (binary system phase diagram) represents the state of a system with two components. In such diagrams, temperature is plotted along the ordinate axis, and the percentage of the components of the mixture is plotted along the abscissa axis (usually it is either a percentage of the total mass (wt.%) or a percentage of the total number of atoms (at.%)). The pressure is usually assumed to be 1 atm. If liquid and solid phases are considered, volume measurements are neglected. In Fig. 2. shows a typical two-phase phase diagram for components A and B using weight or atomic percentage.
Rice. 2.
A letter? the phase of substance A with solute B is indicated, ? means a phase of substance B with substance A dissolved in it, huh? + ? means a mixture of these phases. The letter (from liquid) means the liquid phase, and L+?? and L+? mean liquid phase plus phase or respectively. The lines separating the phases, i.e. the lines on which different phases of a substance can exist, have the following names: solidus - a line on which phases exist simultaneously? or? with L+ phases? and L+? respectively; solvus - a line on which phases coexist simultaneously? And? + ? or? And? + ?, and liquidus is the line on which phase L and phase L+? or L+?.
The point where two liquidus lines intersect is often the point of lowest melting point for all possible combinations of substances A and B and is called the eutectic point. A mixture with a ratio of components at the eutectic point is called a eutectic mixture (or simply eutectic).
Let's consider how a mixture transitions from a liquid state (melt) to a solid state and how the phase diagram helps predict the equilibrium composition of all phases existing at a given temperature. Let's look at Fig. 3.
Rice. 3.
Let us assume that initially the mixture had the composition C M at temperature T1, at temperatures from T1 to T2 there is a liquid phase, and at temperature T2 phases L and? simultaneously exist. The composition of the phase L present is C M, the composition of the phase? there is C ?1. With a further decrease in temperature to T 3, the composition of the liquid changes along the liquidus curve, and the composition of the phase? - along the solidus curve until it intersects with the isotherm (horizontal line) T 3 . Now the composition of phase L is C L, and the composition of phase is C ?2. It should be noted that the composition C ?2 should have not only the substance that has passed into the phase at? at temperature T 3 , but also all the substance that has passed into the phase? at a higher temperature, should have a composition of C ?2. This equalization of compositions must occur by solid-state diffusion of component A into the existing phase?, so that by the time the temperature T 3 is reached, all the substance in the phase? will have the composition C?2. A further decrease in temperature brings us to the eutectic point. Does it have phases? And? exist simultaneously with the liquid phase. At lower temperatures only phases exist? And?. Is a mixture of phases formed? And? composition C E with aggregates? with the initial composition C?3. Then, by keeping this mixture for a long time at a temperature below the eutectic, a solid can be obtained. The resulting solid will consist of two phases. The composition of each phase can be determined at the point of intersection of the isotherm with the corresponding solvus line.
It has just been shown how to determine the composition of each of the phases present. Now consider the problem of determining the amount of substance in each phase. To avoid confusion in Fig. 4. A simple two-phase diagram is shown again. Let us assume that at temperature T 1 the composition of the melt is C M (meaning component B), then at T 2 phase L has the composition C L , and phase? will have the composition C s. Let M L be the mass of a substance in a solid state, and M S be the mass of a substance in a solid state. The condition for conservation of the total mass leads to the following equation
(M L + M S)C M = M L C L + M S C S .
Rice. 4.
It reflects the fact that the total mass of a substance at temperature T1, multiplied by the percentage B, is the total mass of substance B. It is equal to the sum of the masses of substance B existing in the liquid and solid phases at temperature T2. Solving this equation, we get
This expression is known as the "level rule". Using this rule, knowing the initial composition of the melt and its total mass, it is possible to determine the masses of both phases and the amount of substance B in any phase for any section of the two-phase diagram. In exactly the same way we can calculate
In Fig. 5. Another example of melt solidification is given. A decrease in temperature from T 1 to T 2 leads to mixing of phases L and? with the composition C M and C ? . As further cooling occurs, the composition L changes along the liquidus, and the composition? - along the solidus, as described earlier. When the temperature reaches T 3, the composition? will become equal to C M, and, as follows from the level rule, at a temperature below T 3, the liquid phase cannot exist. At temperatures below T4, phases? And? exist as phase aggregates? And?. For example, at a temperature T 5 phase aggregates? will have a composition determined by the intersection of the T5 isotherm and the solvus?. Compound? is determined similarly - by the intersection of the isotherm and solvus?.
Rice. 5.
Sections of a two-phase diagram still called? and?, are areas of solid solubility: in the area? A and B are dissolved. The maximum amount of A that can be dissolved in B at a given temperature depends on the temperature. At eutectic or higher temperatures, rapid fusion of A and B can take place. If the resulting alloy is cooled sharply, then the atoms of A can be “trapped” in the lattice of B. But if the solid solubility at room temperature is much lower (this suggests that at this temperature the approach under consideration is not very suitable), then strong stresses can arise in the alloy, significantly affecting its properties (in the presence of significant stresses, supersaturated solid solutions arise, and the system is not in an equilibrium state, and the diagram provides information only about equilibrium states ). Sometimes, such an effect is desirable, for example, when strengthening steel by quenching to produce martensite. But in microelectronics its result will be destructive. Therefore, doping, i.e. adding additives to silicon before diffusion, is carried out at elevated temperatures in such a way as to prevent surface damage due to excessive alloying. If the amount of doping impurity in the substrate is higher than the solid solubility limit at any temperature, then a second phase appears and the associated deformation.
Rice. 2.3. State diagram of a multicomponent gas.
Unlike a pure substance, for multicomponent systems, a change in volume in the two-phase region is accompanied by a change in pressure (Fig. 2.3, o). To completely evaporate a liquid, it is necessary to continuously lower the pressure and, conversely, to completely condense the gas, it is necessary to continuously increase the pressure. Therefore, the pressure of the point at which vaporization begins for a multicomponent system is higher than the pressure at the point at which condensation begins and when the phase diagram is reconstructed in the coordinates
pressure - temperature curves of evaporation start points and dew points do not coincide. Compared to the phase diagram of a pure substance, the diagram in these coordinates has the shape of a loop (Fig. 2.3,6). The curve of the points of the beginning of vaporization, which is the boundary separating the regions of the liquid and two-phase states of the substance, and the dew point curve, separating the two-phase region from the region of vaporization, are connected at the critical point C. In this case, the critical point is not the point of maximum pressure and temperature at which simultaneously two phases may exist, but, as in the case of a pure substance, at the critical point the density and composition of the phases are the same.
For a multicomponent system, point M with the maximum temperature at which a two-phase state is possible is called cricondenthermoy, a point N s appropriate pressure - cricondenbaroy. Between these points and the critical point there are two regions in which the behavior of the mixture differs from the behavior of the pure substance. During isothermal compression, for example at temperature G, along the line EA, mixture after crossing at a point E the dew point line partially condenses and enters a two-phase state. With a further increase in pressure, the proportion of the liquid phase increases, but only for a certain pressure corresponding to the point D. Subsequent increase in pressure from the point D to the point IN leads to a decrease in the proportion of the liquid phase, and then the mixture returns to the vapor state. Point pressure D, at which the maximum amount of liquid phase is formed is called the pressure of maximum condensation.
Similar phenomena are observed during isobaric heating of a liquid along the line LNGB. Initially, the mixture is in a single-phase liquid state. After crossing the line of points where vaporization begins at the point L a vapor phase appears in the mixture, the amount of which increases to the point N. A subsequent increase in temperature leads to a decrease in the volume of the vapor phase until the substance returns to the liquid state at the point G.
The areas in which condensation and evaporation occur in the direction opposite to the phase transformations of pure matter are called retrograde areas (they are shaded in Fig. 2.3.6). The phenomena occurring in these areas are called retrograde (reverse) evaporation and retrograde (reverse) condensation. These phenomena are widely used in in-field gas treatment processes to select conditions that ensure maximum separation of gas condensate.
Loop-shaped phase diagram (Fig. 2.3, b) is typical for all multicomponent mixtures, but the shape of the loop, the position of the critical point and retrograde regions depend on the composition of the mixture. If the composition of the reservoir mixture is such that the cricondentherm is located to the left of the isotherm corresponding to the reservoir temperature (line ft]), then, as the pressure decreases during field development, this mixture will only be in a single-phase gas state. Mixtures of hydrocarbons of this composition form gas deposits. If the composition of the mixture is such that the reservoir temperature is between the critical temperature and the crycondentherm temperature (line AT^), then such hydrocarbon mixtures form gas condensate fields. In the process of reducing pressure at reservoir temperature, a liquid phase will be released from them - condensate.
For oil fields, the critical point is located to the right of the reservoir temperature isotherm (line GTi). If point G with coordinates corresponding to the initial reservoir pressure and reservoir temperature is located above the line where vaporization begins, then the oil is in a single-phase liquid state and is undersaturated with gas. Only when the pressure drops below the saturation pressure (point D) the gas phase begins to be released from the oil. Oil fields, the composition of the hydrocarbon mixture of which is such that the initial reservoir pressure (point K) is below the saturation pressure, have a gas cap, which is a gas phase accumulated in the upper part of the deposit.